Brane Webs and Magnetic Quivers for SQCD

TL;DR

The paper investigates the classical Higgs branch of 4d $ ext{N}=2$ SQCD using five-brane webs and magnetic quivers, revealing that the Higgs variety is in general a union of hyper-Kähler cones with nilpotent operators appearing in the full Higgs ring. It develops a brane-web/Magnetic-Quiver framework to extract cone components and their intersections, and compares Hilbert-series computations from hyper-Kähler quotients with radicalized Higgs rings, finding perfect agreement once nilpotents are removed. The results provide a detailed map of mesonic and baryonic branches, including their global symmetries via HWGs, and illustrate how radical ideals reconcile brane methods with algebraic geometry, yielding new insights into vacuum structure and phase decomposition in SQCD. The approach offers a scalable alternative to direct computational algebra (e.g., Gröbner bases) for analyzing Higgs branches across a wide range of $(N_c,N_f)$, with potential implications for understanding nilpotent dynamics and multiplicities of cones in these theories.

Abstract

It is widely considered that the classical Higgs branch of 4d $\mathcal{N}=2$ SQCD is a well understood object. However there is no satisfactory understanding of its structure. There are two complications: (1) the Higgs branch chiral ring contains nilpotent elements, as can easily be checked in the case of $\mathrm{SU}(N)$ with 1 flavour. (2) the Higgs branch as a geometric space can in general be decomposed into two cones with nontrivial intersection, the baryonic and mesonic branches. To study the second point in detail we use the recently developed tool of magnetic quivers for five-brane webs, using the fact that the classical Higgs branch for theories with 8 supercharges does not change through dimensional reduction. We compare this approach with the computation of the hyper-Kähler quotient using Hilbert series techniques, finding perfect agreement if nilpotent operators are eliminated by the computation of a so called radical. We study the nature of the nilpotent operators and give conjectures for the Hilbert series of the full Higgs branch, giving new insights into the vacuum structure of 4d $\mathcal{N}=2$ SQCD. In addition we demonstrate the power of the magnetic quiver technique, as it allows us to identify the decomposition into cones, and provides us with the global symmetries of the theory, as a simple alternative to the techniques that were used to date.

Figures (26)

• Figure 1: Summary of the various incarnations of the Higgs branch. The second column contains the coordinate rings, the third column contains the geometric objects. In the second row, the ring contains nilpotent operators, it can be computed from the F-term relations through the hyper-Kähler quotient. Upon taking the radical (see appendix \ref{['AppendixAlgebra']}), the third row contains no nilpotent operators. The associated Higgs branch is an algebraic variety, which is a symplectic singularity or union thereof. As such, it is described by magnetic quivers that can be read from the brane web.
• Figure 2: Depicted are Type IIB brane configurations with supersymmetric gauge theories living on the lightest branes. A different point in their moduli space is depicted with a dashed line. The 3d $\mathcal{N}=4$ gauge theory living on the two D3 branes has 2 (quaternionic) Coulomb branch moduli, while the 5d $\mathcal{N}=1$ gauge theory living on the D5 branes has only one (real) Coulomb branch modulus, as the center of mass is fixed. Hence the 3d theory has a $\mathrm{U}(2)$ gauge group at the origin of the moduli space, while the 5d theory has a $\mathrm{SU}(2)$ gauge group. Both have the Weyl group $S_2=\mathbb{Z}_2$. Decoupling the center of mass of the D3 branes leads to a centerless $\mathrm{U}(2)/\mathrm{U}(1)=\mathrm{SU}(2)/\mathbb{Z}_2$ gauge group. A similar argument holds for 4d $\mathcal{N}=2$ gauge theories living on D4 branes suspended between NS5 branes in Type IIA, which is a T-dual configuration of the above. Because of the logarithmic bending of the NS5 branes the center of mass is fixed and the gauge group is $\mathrm{SU}(2)$. Moving the center of mass in the 4d or 5d case corresponds to changing the asymptotic behaviour Karch:1998yv. The moduli space of the gauge theory corresponds to the brane motions which keep the asymptotic form of the heavy branes intact Karch:1998uy.
• Figure 3: The brane web realisation of $\mathrm{SU}(3)$ SQCD at finite coupling with $N_f=4$ massless flavours at the origin of its moduli space. Horizontal lines correspond to D5 branes, vertical lines correspond to NS5 branes, lines at an angle $\tan(\alpha)=\frac{p}{q}$ from the $x^5$ axis correspond to $(p,q)5$-branes and circles correspond to $[p,q]7$-branes. Note that $(p,q)5$-branes end on $[p,q]7$-branes. The parallel D5 branes are supposed to coincide in space and are drawn slightly appart for clarity.
• Figure 4: Schematic picture of the Coulomb Branch, $\mathcal{M}_C$, and Higgs Branch, $\mathcal{M}_H$, of SCQD with 8 supercharges at finite coupling. The Higgs Branch is a union of two cones, the mesonic cone, $\mathcal{M}_\textrm{M}$, and the baryonic cone, $\mathcal{M}_\textrm{B}$, with non-trivial intersection (line in bold).
• Figure 5: Depiction of different phases in the brane web corresponding to different points in the moduli space of the gauge theory that lives on the web.